4 matched filters and ambiguity functions for radar signals

1

Matched Filters and Ambiguity Functions for

RADAR SignalsPart 1

SOLO HERMELIN

Updated: 01.12.08http://www.solohermelin.com

http://www.solohermelin.com/

2

SOLO Matched Filters and Ambiguity Functions for RADAR Signals

Table of Content

RADAR RF SignalsMaximization of Signal-to-Noise Ratio

Continuous Linear Systems

The Matched FilterThe Matched Filter Approximations

1. Single RF Pulse2. Linear FM Modulated Pulse (Chirp)

Discrete Linear Systems

RADAR SignalsSignal Duration and Bandwidth

Complex Representation of Bandpass SignalsMatched Filter Response to a Band Limited Radar Signal

Matched Filter Response to Phase Coding

Matched Filter Response to its Doppler-Shifted Signal

3

SOLO Matched Filters and Ambiguity Functions for RADAR Signals

Table of Content (continue – 1)

Ambiguity Function for RADAR Signals

Definition of Ambiguity Function

Ambiguity Function Properties

Cuts Through the Ambiguity FunctionAmbiguity as a Measure of Range and Doppler Resolution

Ambiguity Function Close to Origin

Ambiguity Function for Single RF Pulse

Ambiguity Function for Linear FM Modulation Pulse

Ambiguity Function for a Coherent Pulse TrainAmbiguity Function Examples (Rihaczek, A.W.,

“Principles of High Resolution Radar”)

References

AMBIGUITY

FUNCTIONS

4

SOLO

The transmitted RADAR RF Signal is:

( ) ( ) ( )[ ]ttftEtEt 0000 2cos ϕπ +=E0 – amplitude of the signal

f0 – RF frequency of the signal φ0 –phase of the signal (possible modulated)

The returned signal is delayed by the time that takes to signal to reach the target and toreturn back to the receiver. Since the electromagnetic waves travel with the speed of lightc (much greater then RADAR andTarget velocities), the received signal is delayed by

c

RRtd

21 +≅

The received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos

To retrieve the range (and range-rate) information from the received signal thetransmitted signal must be modulated in Amplitude or/and Frequency or/and Phase.

ά < 1 represents the attenuation of the signal

RADAR Signal ProcessingRADAR RF Signals

5

SOLO

The received signal is:

( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos

( ) ( ) tRRtRtRRtR ⋅+=⋅+= 222111 &

We want to compute the delay time td due to the time td1 it takes the EM-wave to reachthe target at a distance R1 (at t=0), from the transmitter, and to the time td2 it takes the EM-wave to return to the receiver, at a distance R2 (at t=0) from the target. 21 ddd ttt +=

According to the Special Theory of Relativitythe EM wave will travel with a constant velocity c (independent of the relative velocities ).21& RR

The EM wave that reached the target at time t was send at td1 ,therefore

( ) ( ) 111111 ddd tcttRRttR ⋅=−⋅+=− ( )1

111 Rc

tRRttd

+⋅+=

In the same way the EM wave received from the target at time t was reflected at td2 , therefore

( ) ( ) 222222 ddd tcttRRttR ⋅=−⋅+=− ( )2

222 Rc

tRRttd

+⋅+=

RADAR Signal Processing

6

SOLO

The received signal is:

( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos

21 ddd ttt += ( )1

111 Rc

tRRttd

+⋅+= ( )

2

222 Rc

tRRttd

+⋅+=

( ) ( )2

22

1

1121 Rc

tRR

Rc

tRRtttttttt ddd

+⋅+−

+⋅+−=−−=−

+

−+−+

+

−+−=−

2

2

2

2

1

1

1

1

2

1

2

1

Rc

Rt

Rc

Rc

Rc

Rt

Rc

Rctt d

From which:

or:

Since in most applications we canapproximate where they appear in the arguments of E0 (t-td), φ (t-td),however, because f0 is of order of 109 Hz=1 GHz, in radar applications, we must use:

cRR <<21,

1,2

2

1

1 ≈+−

+−

Rc

Rc

Rc

Rc

( )

−⋅

++

−⋅

+=

−⋅

−+

−⋅

−⋅≈− 2

.

201

.

1022

011

00 2

1

2

1

2

121

2

121

21

D

RalongFreqDoppler

DD

RalongFreqDoppler

Dd ttffttffc

Rt

c

Rf

c

Rt

c

Rfttf

( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−= ˆˆˆ2cosˆ00 ϕπα

where 212

21

1212

021

01ˆˆˆ,,,ˆˆˆ,

2ˆ,2ˆ

dddddDDDDD tttc

Rt

c

Rtfff

c

Rff

c

Rff +=≈≈+=−≈−≈

Finally

Matched Filters in RADAR Systems

Doppler Effect

7

SOLO

The received signal model:

( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−= ϕπα 00 2cos

Matched Filters in RADAR Systems

Delayed by two-way trip time

Scaled downAmplitude Possible phase

modulated

CorruptedBy noise

Dopplereffect

We want to estimate:

• delay td range c td/2

• amplitude reduction α

• Doppler frequency fD

• noise power n (relative to signal power)

• phase modulation φ

8

Matched Filters in RADAR SystemsSOLO

α MV R

EVTarget

Transmitter &Receiver

The transmitted RADAR RF Signal is:

( ) ( ) ( )[ ]ttftEtEt θπ += 00 2cos

( )c

tRtd

02≅

Since the received signal preserve the envelope shape of the known transmitted signalwe want to design a Matched Filter that will distinguish the signal from the receiver noise.

( ) ( )λ

λ0

/

00 22 0 tR

fc

tRf

fc

D

−=−≅

=

the received signal is: ( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−+−≈ θπα 00 2cos

Scaled DownIn Amplitude Two-Way

Delay

Possible Phase Modulation

DopplerFrequency

For R1 = R2 = R we obtain that

Return to Table of Content

9

Matched Filters for RADAR Signals

( ) ( ) ( )tntstv ii += Linear Filter

( )thopt

( ) ( ) ( )tntsty oo +=

SOLO

Maximization of Signal-to-Noise Ratio

Consider the problem of choosing a linear time-invariant filter hopt (t) that maximizesthe output signal-to-noise ratio at a predefined time t0.

The input waveform is: ( ) ( ) ( )tntstv ii +=

( )tsi - a known signal component

( )tni - noise (stationary random process) component

The output waveform is: ( ) ( ) ( )tntsty oo +=

Assume that the linear filter has a finite time memory T, then

( ) ( ) ( )∫ −=T

iopto dtshts0

00 τττ ( ) ( ) ( )∫ −=T

iopto dtnhtn0

00 τττ

The signal-to-noise ratio is defined as:( )( )02

02

tn

ts

N

S

o

o=

To find hopt (t) a variational technique is applied, by defining a non-optimal filter

( ) ( ) ( )tgthth opt ε+= ( ) ( ) 00

0 =−∫T

i dtsg τττwith: and ε any real.

( )02 tno - the mean square value of ( )0tno

Continuous Linear Systems

10



( ) ( )tgthopt ε+( ) ( ) ( )tntsty oo ''' +=

SOLOMaximization of Signal-to-Noise Ratio

The output signal s’o (t) and noise n’o (t) at time t0 are:

( ) ( ) ( )[ ] ( )

( ) ( ) ( ) ( ) ( )00

0

0

0

0

0

00'

tsdtsgdtsh

dtsghts

o

T

i

T

iopt

T

iopto

=−+−=

−+=

∫∫

∫

τττετττ

τττετ

( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )∫

∫∫∫

−+=

−+−=−+=

T

io

T

i

T

iopt

T

iopto

dtngtn

dtngdtnhdtnghtn

0

00

0

0

0

0

0

00'

τττε

τττεττττττετ

( )[ ] ( )[ ] ( ) ( ) ( ) ( ) ( )2

0

02

0

002

02

0 2'

−+−+= ∫∫

T

i

T

iooo dtngdtngtntntn τττετττε

By the definition of the optimal filter ( )[ ] ( )[ ]202

0' tntn oo ≥

Therefore ( ) ( ) ( ) ( ) ( ) 02

2

0

02

0

00 ≥

−+− ∫∫

T

i

T

io dtngdtngtn τττετττε

Continuous Linear Systems (continue – 1(

11





This inequality is satisfied for all values of ε if and only if the first term vanishes

( ) ( ) ( ) ( ) ( ) 02

2

0

02

0

00 ≥

−+− ∫∫

T

i

T

io dtngdtngtn τττετττε

( ) ( ) 00

0 =−∫T

i dtsg τττ

( ) ( ) ( ) 020

00 =−∫T

io dtngtn τττ

Using we obtain:( ) ( ) ( )∫ −=T

iopto dtnhtn0

00 τττ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00 0

00

0 0

00 =−−=−− ∫ ∫∫ ∫T T

iiopt

T T

iiopt ddtntnhgddtntnhg στστστστστστ

where is the Autocorrelation Function of the input noise.( ) ( ) ( )στστ −−=− 00: tntnR iinn ii


12





Therefore the optimality condition is:

( ) ( ) ( ) 00 0

=

−∫ ∫ τσστστ ddRhg

T T

nnopt ii

( ) ( ) 00

0 =−∫T

i dtsg τττComparing with the condition:

we obtain:

( ) ( ) ( ) TtskdRh i

T

nnopt ii≤≤−=−∫ ττσστσ 00

0

k is obtained using:

( ) ( ) ( ) ( ) ( ) ( ) ( )k

tnddRhh

kdtshts o

T T

nnoptopt

T

iopto ii

02

0 00

00

1 =−=−= ∫ ∫∫ στστσττττ( )

( )00

2

ts

tnk

o

o=

For T → ∞ we can take the Fourier Transfer of the result:

( ) ( ) ( )( ) ( )[ ]τσστσ −=

−∫∞→ 0

0

02

0

lim tsts

tndRh i

o

oT

nnoptT ii

FF


13

Matched Filters for RADAR SignalsSOLO

Maximization of Signal-to-Noise Ratio

( ) ( ) ( )( ) ( )[ ]τσστσ −=

−∫

∞

00

02

0

tsts

tndRh i

o

onnopt ii

FF


( )thopt

( ) ( ) ( )tntsty oo +=

( ) ( ) ( )( ) ( ) 0*

0

02

tji

o

onnopt eS

ts

tnH

ii

ωωωω −=Φ ( ) ( )( )

( )( )ω

ωωω

iinn

tji

o

oopt

eS

ts

tnH

Φ=

− 0*

0

02



14


( )tsi

t

T0mt

SOLO

The Matched Filter

Assume that the two-sided noise spectrum density is of a white noise, i.e.

( ) ( )στδστ −=−20N

Riinn

( ) ( )[ ]20N

Riiii nnnn ==Φ τω F

then

( ) ( ) 0*2 tji

oopt eS

N

kH ωωω −= ( ) ( ) Tttts

N

kth i

oopt ≤≤−= 0

20

( )tsi

t

t

( )tsi −T0

0T−

mt

( )tsi

t

t

t

( )tsi −

( ) ( ) TttttsN

kth mmiopt ≤≤−= ,0

2

0

T0

0T−

0 mtmtT −

mt

The optimal filter, that maximizes the Signal-to-Noise Ratio for a white noise iscalled a Matched Filter of the knownSignal si (t).

We can see that for a known inputsignal of finite duration T the optimal Matched Filter is also of finite duration T.

15


( ) ( ) ( ) ( ) 02

00

2 tjiiopt eS

N

kSHS ωωωωω −==

SOLO

The Matched Filter

The signal and the noise at the output of the matched filter are found as follows:

then ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫+∞

∞−

+∞

∞−

−−+∞

∞−

−

==

πωω

πωω ωωω

2

2

2

20*2 d

dvevseSN

kdeS

N

kts vj

ittj

io

ttji

oo

m

( ) ( ) 0*2 tji

oopt eS

N

kH ωωω −=

( ) ( ) ( ) ( ) ( )∫∫ ∫+∞

∞−

+∞

∞−

+∞

∞−

−− +−== dvttvsvsN

kdv

deSvs

N

kii

o

vttjii

o0

* 2

2

20

πωω ω

The Autocorrelation Function of the input signal is defined as: ( ) ( ) ( )∫+∞

∞−

−= dvvsvsR iiss iiττ :

therefore: ( ) ( )02

ttRN

kts

iisso

o −=

( ) ( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

∗

=Φ= ωω

πωωωω

πd

NS

N

kdHHtn o

io

optnnopto ii 2

2

2

1

2

1 22

2( ) ( ) mtj

i

o

opt eSN

kH ωωω −= *2

( ) ( ) ( )[ ]∫+∞

∞−

== dvvsN

kR

N

kts i

oss

oo ii

20

20

2

16


The Matched Filter

therefore:

( ) ( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

∗

=Φ= ωω

πωωωω

πd

NS

N

kdHHtn o

io

optnnopto ii 2

2

2

1

2

1 22

2

( ) ( ) ( )[ ]∫+∞

∞−

== dvvsN

kR

N

kts i

oss

oo ii

20

20

2

( )[ ]( )

( )[ ]

( )

( )[ ]

( )∫

∫

∫

∫∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

=

==

πωω

πωω 2222

2

2

2

2

2

22

2

2

2

20

dS

N

dvvs

dNS

Nk

dvvsNk

tn

ts

N

S

io

i

oi

o

io

o

o

Max

Since by Parseval’s relation: (E – input signal energy)( )[ ] ( ) Ed

Sdvvs ii == ∫∫+∞

∞−

+∞

∞−πωω2

22

( )[ ]( ) oo

o

Max N

E

tn

ts

N

S 22

20 ==

We have: ( ) ( ) ( )∫+∞

∞−

+−= dvttvsvsN

kts ii

oto 0

20

Independent of signal waveform

17


( ) ( )( ) ( )

≤≤−== −∗

Ttttsth

eSH tj

00

0ωωω

SOLO

The Matched Filter (Summary(

s (t) - Signal waveform

S (ω) - Signal spectral density

h (t) - Filter impulse response

H (ω) - Filter transfer function

t0 - Time filter output is sampled (for Radar signals this is the time the received returned signal is expected to arrive)

n (t) - noise

N (ω) - Noise spectral density

Matched Filter is a linear time-invariant filter hopt (t) that maximizesthe output signal-to-noise ratio at a predefined time t0, for a given signal s (t(.

The Matched Filter output is:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) 0

00

tjo eSSHSS

dttssdthsts

ωωωωωω

ττττττ

−∗

+∞

∞−

+∞

∞−

⋅=⋅=

+−=−= ∫∫


18


Matched Filter Output for White Noise Spectrum

s (t) - Signal waveform with energy E

S (ω) - Signal spectral density

h (t) - Filter impulse response

H (ω) - Filter transfer function

( ) ( ) ( ) ( ) ( ) ( ) ( )0* 0

2

1

2

1ttRdeSSdeSHts ss

ttjtjo −=== ∫∫

+∞

∞−

−+∞

∞−

ωωωπ

ωωωπ

ωω

( ) ( ) ( ) ( )0*'

0

2

ss

TheoremsParsevalT

RdSSdttsE === ∫∫+∞

∞−

ωωω

so (t) - Filter output signal

N (ω) - Noise spectral density η/2

Rnn (τ) - Noise Autocorrelation Function η/2 δ (τ) ( ) ( ) ( )∫−

∞→+=

T

TT

nn dttntnT

R ττ 1lim

Rss (τ) - Signal Autocorrelation Function ( ) ( ) ( )∫−

∞→+=

T

TT

ss dttstsT

R ττ 1lim

S/N - Output Power signal-to-noise ratio E/(η/2)

t0 - Time filter output is sampled (for Radar signals this is the time the received returned signal is expected to arrive)


19


The Matched Filter Approximations

1. Single RF Pulse

( )( )

>

≤≤−=

2/0

2/2/cos 0

p

pp

itt

ttttAts

ω

pt - pulse width

( ) ( )

( )

( )

( )

( )

−

−

++

+

=

= ∫−

−

2

2sin

2

2sin

2

cos

0

0

0

0

2/

2/

0

p

p

p

p

p

t

t

tji

t

t

t

t

tA

dtetAjSp

p

ωω

ωω

ωω

ωω

ωω ω

Fourier Transform

0ω - carrier frequency

We found: ( ) ( ) ( ) ( )∫+∞

∞−

−=−= dvtvsvsN

kttR

N

kts ii

otss

oto ii

2200

0

therefore:

( ) ( )( )

( )

( )[ ]

( )[ ]

( ) ( )[ ]

( ) ( )[ ]

( )( ) ( ) ttt

N

tkAttt

N

kA

tttvttt

tttvttt

N

kA

ttdvtvt

ttdvtvt

N

kA

ttdvtvAvA

ttdvtvAvA

N

ktR

N

kts

po

pp

o

tt

ptt

tp

pt

ttp

o

p

tt

t

p

t

tt

ott

t

p

p

t

tt

oss

oto

p

p

p

p

p

p

p

p

p

p

p

p

p

ii

0

2

0

21

1

2/

2/00

0

2/

2/00

02

2/

2/

00

2/

2/

002

2/

2/

00

2/

2/

00

0

cos/1cos

02sin2

1cos

02sin2

1cos

02coscos

02coscos

0coscos

0coscos22

02

0

ωωω

ωω

ωω

ω

ωω

ωω

ωω

ωω

ω

−=−≈

<<−−++

<<−+−=

<<−−+

<<−+

=

<<−−

<<−

==

<<−

+−

−

+

−

−

+

−

−

=

∫

∫

∫

∫


( )thopt

( ) ( ) ( )tntsty oo +=

20



1. Single RF Pulse (continue – 1(

( )( )

>

≤≤−=

2/0

2/2/cos 0

p

pp

itt

ttttAts

ω

pt - pulse width

( ) ( )( )

( )

( )

( )0

0

2

2sin

2

2sin

2 0

0

0

0

*

tj

p

p

p

p

p

tjiMF

et

t

t

t

tA

ejSjS

ω

ω

ωω

ωω

ωω

ωωωω

−

−

−

−

++

+

=

=0ω - carrier frequency

We obtained:

( ) ( )

≥

<−=

=

p

ppo

p

to

tt

tttttN

tkA

ts

0

cos/1 0

2

00

ω


( )thopt

( ) ( ) ( )tntsty oo +=

t2

τ2

τ−

( )tso

0

2

N

Ak τ

ττ−

0=mt


21



1. Single RF Pulse (continue – 2(

( )( )

>

≤≤−=

2/0

2/2/cos 0

p

pp

itt

ttttAts

ω

pt - pulse width

( ) ( )( )

( )

( )

( )0

0

2

2sin

2

2sin

2 0

0

0

0

*

tj

p

p

p

p

p

tjiMF

et

t

t

t

tA

ejSjS

ω

ω

ωω

ωω

ωω

ωωωω

−

−

−

−

++

+

=

=

0ω - carrier frequency

We obtained:


22

SOLO

2. Linear FM Modulated Pulse (Chirp)

( )222

cos2

0pp

i

tt

tttAts ≤≤−

+= µω

The Fourier Transform is:

( ) [ ]

( ) ( )∫∫

∫

−−

−

++−+

+−=

−

+=

2/

2/

2

0

2/

2/

2

0

2/

2/

2

0

2exp

2

1

2exp

2

1

exp2

cos

p

p

p

p

p

p

t

t

t

t

t

t

i

dtt

tjAdtt

tjA

dttjt

tAS

µωωµωω

ωµωω

∫∫−−

++−

++

−−

−−=2/

2/

2

0

2

0

2/

2/

2

0

2

0

2exp

2exp

22exp

2exp

2

p

p

p

p

t

t

t

t

dttjjA

dttjjA

µωωµ

µωω

µωωµ

µωω

Change variables: xt =

−−

µωω

πµ 0 yt =

++

µωω

πµ 0

( ) ∫∫−−

−

++

−−=

2

1

2

1

2exp

2exp

22exp

2exp

2

22

02

2

0

Y

Y

X

X

i dty

jjA

dtx

jjA

Sπ

µωωπ

µωωω

−−=

−+=µ

ωωπµ

µωω

πµ 0

20

1 2&

2pp t

Xt

X

+−=

++=µ

ωωπµ

µωω

πµ 0

20

1 2&

2pp t

Yt

Y

Define: ( )fntf p ∆=−=∆ πωωµ

π2

2&

2

1: 0


23

SOLO

2. Linear FM Modulated Pulse (continue – 1)


( ) ( ) ( )∫∫

−−

−

++

−−=2

1

2

12

exp2

exp22

exp2

exp2

220

220

Y

Y

X

X

i dty

jjA

dtx

jjA

Sπ

µωωπ

µωωω

The first part gives the spectrum around ω = ω0, and the second part around ω = -ω0 :

where: are Fresnel Integrals,

which have the properties:

( ) ( ) ∫∫ ==UU

dzz

USdzz

UC0

2

0

2

2sin&

2cos

ππ

( ) ( ) ( ) ( )USUSUCUC −=−−=− &

( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) −+ ++−=−+−

−+

+++

−−=

ωωωωµωω

µπ

µωω

µπω

002211

2

0

2211

2

0

2exp

2

2exp

2

ii

i

SSYSjYCYSjYCjA

XSjXCXSjXCjA

S


( )222

cos2

0pp

i

tt

tttAts ≤≤−

+= µω

ωωωπωµπ

∆=−∆=∆=∆2

:&2:2

1: 0

nftf p

24

SOLO Fresnel Integrals

Augustin Jean Fresnel1788-1827

Define Fresnel Integrals

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∫ ∑

∑∫∞

=

+

∞

=

+

+−=

=

++−=

=

α

α

ααπα

ααπα

0 0

142

0

34

0

2

!2141

2sin:

!12341

2cos:

n

nn

n

nn

nn

xdS

nn

xdC

( ) ( )ααααπα

SjCdj +=

∫

0

2

2exp

( ) ( ) 5.0±=∞±=∞± SC

( ) ( ) ( ) ( )USUSUCUC −=−−=− &

The Cornu Spiral is defined as the plot of S (u) versus C (u)

duuSd

duuCd

=

=

2

2

2sin

2cos

π

π

( ) ( ) duSdCd =+ 22

Therefore u may be thought as measuring arc length along the spiral.

25

SOLO



Define:

( ) ( ) ( )[ ] ( ) ( )[ ]{ }221

2

210 2XSXSXCXC

AS i +++=−

+ µπωωAmplitude Term:

Square Law Phase Term: ( ) ( )µωωω2

2

01

−−=Φ

Residual Phase Term: ( ) ( ) ( )( ) ( ) 4

1tan5.05.0

5.05.0tantan 111

21

211

2

πωτ

==++→

++=Φ −−

>>∆−

f

XCXC

XSXS

( ) ( )ntfXn

tfX pp −∆=+∆= 1

2&1

2 21

( )ω2Φ( )+

− ωω 0iS( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) −+ ++−=−+−

−+

+++

−−=

ωωωωµωω

µπ

µωω

µπω

002211

2

0

2211

2

0

2exp

2

2exp

2

ii

i

SSYSjYCYSjYCjA

XSjXCXSjXCjA

S


( )222

cos2

0pp

i

tt

tttAts ≤≤−

+= µω

ωωωπωµπ

∆=−∆=∆=∆2

:&2:2

1: 0

nftf p

26

SOLO



( )222

cos2

0pp

i

tt

tttAts ≤≤−

+= µω

ωωωπωµπ

∆=−∆=∆=∆2

:&2:2

1: 0

nftf p

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) −+−

−−

++−=−+−

−+

+++

−−==

ωωωωµωω

µπ

µωω

µπωω

ω

ωω

002211

20

2211

20*

0

00

2exp

2

2exp

2

MFMFtj

tjtjiMF

SSeYSjYCYSjYCjA

eXSjXCXSjXCjA

eSS

27

SOLO



( )222

cos2

0pp

i

tt

tttAts ≤≤−

+= µω ωωωπωµ

π∆=−∆=∆=∆2

:&2:2

1: 0

nftf p

The Matched Filter output is given by: ( ) ( ) ( ) ( )∫+∞

∞−

−=−= dvtvsvsN

kttR

N

kts ii

otss

oto ii

2200

0

( )( ) ( )

( ) ( )

<<−

−+−

+

<<

−+−

+

=

∫

∫+

−

+

−

2/

2/

2

0

2

0

2/

2/

2

0

2

0

02

cos2

cos

02

cos2

cos2

0 p

p

p

p

tt

t

p

t

tt

p

oto

ttdvtv

tvv

v

ttdvtv

tvv

v

N

kts

µωµω

µωµω

We discard the double frequency term, whose contribution to the value of integral is small for large ω0,

( )( )

( )

<<−

−++−+

−+

<<

−++−+

−+

=

∫

∫+

−

+

−

2/

2/

22

0

2

0

2/

2/

22

0

2

02

02

222cos

2cos

02

222cos

2cos

0 p

p

p

p

tt

t

p

t

tt

p

oto

ttdvtvtv

tvt

tvt

ttdvtvtv

tvt

tvt

N

Akts

µµµωµµω

µµµωµµω

( )

( )

<<−−+

−++−+

+−

<<−+

−++−+

+−

=+

−

+

−

+

−

+

−

0242

222

2sin2

sin

0242

2

222sin

2sin

2/

2/

0

22

0

2/

2/

2

0

2/

2/

0

22

0

2/

2/

2

0

2

tttv

tvtvtv

t

tvt

t

tttv

tvtvtv

t

tvt

t

N

Ak

p

tt

t

tt

t

p

t

tt

t

tt

o p

p

p

p

p

p

p

p

µµω

µµµω

µ

µµω

µµω

µµµω

µ

µµω

Expanding the integrand trigonometrically

28

SOLO




( )222

cos2

0pp

i

tt

tttAts ≤≤−

+= µω

ωωωπωµπ

∆=−∆=∆=∆2

:&2:2

1: 0

nftf p

The Matched Filter output is given by:

( )

<<−

+−

<<

+−

≈+

−

+

−

02

sin

02

sin

2/

2/

2

0

2/

2/

2

02

0

tttvt

t

tttvt

t

tN

Akts

p

tt

t

p

t

tt

oto

p

p

p

p

µµω

µµω

µ

( )

( )

<<−

−−−

++−

<<

−+−−

+−

=0

22sin2/

2sin

02/2

sin22

sin

2

0

2

0

2

0

2

02

ttttt

ttttt

t

tttttt

tttt

t

tN

Ak

pp

p

ppp

o µµωµµω

µµωµµω

µ

( ) ( )

( ) ( )

( )( )

( )( )

>

<

−

−

−=

<<−

+

<<

−

=

p

p

pp

pp

po

p

pp

pp

o

tt

ttt

tttt

tttt

ttN

tAk

tttttt

tttttt

tN

Ak

0

cos

/12

/12

sin

/12

0cos2

sin2

0cos2

sin20

2

0

02 ωµ

µ

ωµ

ωµ

µ

29

SOLO




( )222

cos2

0pp

i

tt

tttAts ≤≤−

+= µω

ωωωπωµπ

∆=−∆=∆=∆2

:&2:2

1: 0

nftf p

The Matched Filter output is given by:

( )( )

( )

( )( )

>

<

−

−

−≈

p

p

pp

pp

po

p

to

tt

ttttt

tt

tttt

ttN

tAk

ts

0

cos/1

2

/12

sin

/1 0

2

0

ωµ

µ

o

p

N

tAk 2

ptt

µπ2=∆

1>>ptµ

30

SOLO




( )222

cos2

0pp

i

tt

tttAts ≤≤−

+= µω

ωωωπωµπ

∆=−∆=∆=∆2

:&2:2

1: 0

nftf p

31


( ) ( ) ( )tntstv ii +=Linear Filter

( )Tnhopt

( ) ( ) ( )tntsty oo +=

( ) ( ) ( )TnnTnsTnv ii +=

T T

( ) ( ) ( )TnnTnsTny oo +=


Consider the problem of choosing a discrete linear time-invariant filter hopt (n T) that Maximizes the discrete output signal-to-noise ratio at a predefined time mT.

The input waveform is: ( ) ( ) ( )tntstv ii +=

( )tsi - a known signal component

( )tni - noise (stationary random process) component

The output waveform is: ( ) ( ) ( )TnnTnsTny oo +=

The signal-to-noise ratio at discrete time mT is defined as:( )( )Tmn

Tms

N

S

o

o

2

2

=

( )Tmno2

- the mean square value of ( )Tmno

Discrete Linear Systems

The input and output of the discrete linear filter are synchronous discretized witha constant time period T. S (z) is the Z -transform of the discrete signal input si (nT)

We have: ( ) ( ) ( )∫+

−

=σ

σ

ωωω ωσ

deeeTns TjTjTj

o HS2

1

( ){ } ( ) ( )∫+

−

=σ

σ

ω ωωσ

deTnnE Tjo

22

2

1HN

32


( ) ( ) ( )tntstv ii +=Linear Filter

( )Tnhopt

( ) ( ) ( )tntsty oo +=

( ) ( ) ( )TnnTnsTnv ii +=

T T

( ) ( ) ( )TnnTnsTny oo +=


Consider the problem of choosing a discrete linear time-invariant filter hopt (n T) that Maximizes the discrete output signal-to-noise ratio at a predefined time mT.

Like in the continuous case the optimal H (z) is:

( )( )

( ) ( )

( ) ( )∫

∫+

−

+

−==

σ

σ

ω

σ

σ

ω

ωωσ

ωωσ

de

de

Tmn

Tms

N

S

Tj

Tj

i

o

o

2

2

2

2

2

1

2

1

HN

HS

Discrete Linear Systems (continue – 1(

If N (ω) = N0 we have:

( ) ( ) ( )∫+

−

=σ

σ

ωωω ωσ

deeeTns TjTjTj

o HS i2

1

( ){ } ( ) ( )∫+

−

=σ

σ

ω ωωσ

deTnnE Tj

o

22

2

1HN

( ) ( )( )

mTjTj

iTj ee

ke ωω

ω

ω−=

N

SH

( ) [ ] [ ]nmsN

knhz

zN

kz i

m

i −=⇔

= −

00

1SH


33

RADAR SignalsSOLO

Waveforms

( ) ( ) ( )[ ]tttats θω += 0cos

a (t) – nonnegative function that represents any amplitude modulation (AM)

θ (t) – phase angle associated with any frequency modulation (FM)

ω0 – nominal carrier angular frequency ω0 = 2 π f0

f0 – nominal carrier frequency

Transmitted Signal

( ) ( ) ( )[ ]{ }ttjtats θω += 0exp

Phasor (complex) Transmitted Signal

34

RADAR SignalsSOLO

Quadrature Form( ) ( ) ( )[ ]

( ) ( )[ ] ( ) ( ) ( )[ ] ( )tttattta

tttats

00

0

sinsincoscos

cos

ωθωθθω

−=+=

where: ( ) ( ) ( )[ ]( ) ( ) ( )[ ]ttats

ttats

Q

I

θθ

sin

cos

==

( ) ( ) ( ) ( ) ( )ttsttsts QI 00 sincos ωω −=

One other form: ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]tjtjtjtj eeta

tttats θωθωθω −−+ +=+= 00

2cos 0

( ) ( ) ( )[ ]tjtj etgetgts 00 *

2

1 ωω −+= ( ) ( ) ( ) ( ) ( )tjQI etatsjtstg θ=+=:

Envelope of the signal

( ) ( ) tjetgts 0ω=

Phasor (complex) Transmitted Signal

35

RADAR SignalsSOLO

Spectrum

Define the Fourier Transfer F

( ) ( ){ } ( ) ( )∫+∞

∞−

−== dttjtstsS ωω exp:F ( ) ( ){ } ( ) ( )∫+∞

∞−

==πωωωω2

exp:d

tjSSts -1F

( ) ( ) ( )[ ]tjtj etgetgts 00 *

2

1 ωω −+= ( ) ( ) ( )[ ]0*

02

1 ωωωωω −−+−= GGS-1FF

-1FF

( ) ( ) ( ) ( ) ( )tjQI etatsjtstg θ=+=:

( ) ( ) ( )[ ]tttats θω += 0cosInverse Fourier Transfer F -1

Envelope of the signalWe defined:

36

RADAR SignalsSOLO

Energy ( ) ( ) ( )[ ]tttats θω += 0cos

( ) ( ) ( )[ ]{ } ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

≈++== dttadttttadttsEs2

022

2

122cos1

2

1: θω

Parseval’s Formula

Proof:

( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

= ωωωπ

dFFdttftf 2*

12*

1 2

1

( ) ( ) ( )∫+∞

∞−

−= dttjtfF ωω exp11

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∫∫+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

=−=−=πωωω

πωωω

πωωω

22exp

2exp 2

*

112*

2*

12*

1

dFF

ddttjtfFdt

dtjFtfdttftf

( ) ( ) ( )∫+∞

∞−

−=πωωω2

exp*

2

*

2

dtjFtf

If s (t) is real, than s (t) = s*(t) and

( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

=== ωωπ

dSdttsdttsEs

222

2

1:

37

RADAR SignalsSOLO

Energy (continue – 1) ( ) ( ) ( )[ ]tttats θω += 0cos

( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

=== ωωπ

dSdttsdttsEs

222

2

1:

( ) ( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

−−−+−−−+

−−−−+−−=

−−+−−−+−=

−

−−

00

0000

0

*

0

*2

00

0

*

00

*

0

00

*

0

*

0

*

4

1

4

1

ϕϕ

ϕϕϕϕ

ωωωωωωωωωωωωωωωω

ωωωωωωωωωω

jj

jjjj

eGGeGG

GGGG

eGeGeGeGSS

For finite band (W << ω0 ) signals (see Figure)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )∫∫∫

∫∫∞+

∞−

∞+

∞−

∞+

∞−

+∞

∞−

−+∞

∞−

=−−−−=−−

≈−−−=−−−

ωωωωωωωωωωωωω

ωωωωωωωωωω ϕϕ

dGGdGGdGG

deGGdeGG jj

*

0

*

00

*

0

2

0

*

0

*2

00 000

( ) ( ) gs EdGdSE 22

1

2

1

2

1:

22 =≈= ∫∫+∞

∞−

+∞

∞−

ωωπ

ωωπ


38

Signals

( ) ( )∫+∞

∞−

= fdefSts tfi π2

SOLO

Signal Duration and Bandwidth

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∫∫ ∫

∫ ∫∫ ∫∫∞+

∞−

∞+

∞−

∞+

∞−

−

∞+

∞−

∞+

∞−

−∞+

∞−

∞+

∞−

∞+

∞−

=

=

=

=

dffSfSdfdesfS

dfdefSsdfdefSsdss

tfi

tfitfi

ττ

τττττττ

π

ππ

2

22

( ) ( )∫+∞

∞−

= fdefSts tfi π2 ( ) ( ) ( )∫+∞

∞−

== fdefSfitd

tsdts tfi ππ 22'

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )∫∫ ∫

∫ ∫∫ ∫∫∞+

∞−

∞+

∞−

∞+

∞−

−

+∞

∞−

+∞

∞−

−+∞

∞−

+∞

∞−

−+∞

∞−

=

−=

−=

−=

dffSfSfdfdesfSfi

dfdesfSfidfdefSfsidss

tfi

tfitfi

222

22

2'2

'2'2''

πττπ

ττπττπτττ

π

ππ

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSds 22 ττ

Parseval Theorem

From

From

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSfdtts2222

4' π

39

Signals

( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )∫

∫

∫

∫ ∫

∫

∫ ∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−

−

∞+

∞−

+∞

∞−

+∞

∞−

−

∞+

∞−

+∞

∞−∞+

∞−

+∞

∞− =====dffS

fdfdfSd

fSi

dffS

fdtdetstfS

dffS

tdfdefStst

dffS

tdtstst

tdts

tdtst

t

fifi

22

2

2

2

22

2

2:

πππ

SOLO

Signal Duration and Bandwidth (continue – 1)

( ) ( )∫+∞

∞−

−= tdetsfS tfi π2 ( ) ( )∫+∞

∞−

= fdefSts tfi π2Fourier

( ) ( )∫+∞

∞−

−−= tdetstifd

fSd tfi ππ 22( ) ( )∫

+∞

∞−

= fdefSfitd

tsd tfi ππ 22

( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )∫

∫

∫

∫ ∫

∫

∫ ∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

−=

====tdts

tdtd

tsdtsi

tdts

tdfdefSfts

tdts

fdtdetsfSf

tdts

fdfSfSf

fdfS

fdfSf

f

fifi

22

2

2

2

22

2 2222

:

ππ ππππ

40

Signals

( ) ( ) ( ) ( ) ( )∫∫∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

=≤

dffSfdttstdttsdttstdtts

222222

2

2 4'4

1 π

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSdts22 τ

SOLO


0&0 == ftChange time and frequency scale to get

From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

≤ dttgdttfdttgtf22

Choose ( ) ( ) ( ) ( ) ( )tstd

tsdtgtsttf ':& ===

( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

≤ dttsdttstdttstst22

''we obtain

( ) ( )∫+∞

∞−

dttstst 'Integrate by parts( )

=+=

→

==

sv

dtstsdu

dtsdv

stu '

'

( ) ( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

∞+

∞−

+∞

∞−

−−= dttststdttsstdttstst '' 2

0

2

( ) ( ) ( )∫∫

+∞

∞−

+∞

∞−

−= dttsdttstst 2

2

1'

( ) ( )∫∫+∞

∞−

+ ∞

∞−

= dffSfdtts2222

4' π

( )

( )

( )

( )

( )

( )

( )

( )∫

∫

∫

∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞− =≤dffS

dffSf

dtts

dttst

dtts

dffSf

dtts

dttst

2

222

2

2

2

222

2

244

4

1ππ

assume ( ) 0lim =→∞

tstt

41

SignalsSOLO


( )

( )

( )

( )

( )

( )

22

2

222

2

24

4

1

ft

dffS

dffSf

dtts

dttst

∆

∞+

∞−

+∞

∞−

∆

∞+

∞−

+∞

∞−

≤

∫

∫

∫

∫ π

Finally we obtain ( ) ( )ft ∆∆≤2

1


Since Schwarz Inequality: becomes an equalityif and only if g (t) = k f (t), then for:

( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−


( ) ( ) ( ) ( )tftsteAttd

sdtgeAts tt ααα αα 222:

22

−=−=−==⇒= −−

we have ( ) ( )ft ∆∆=2

1

42

Signals

t

t∆2

t

( ) 2ts

ff

f∆2( ) 2fS

SOLO

Signal Duration and Bandwidth – Summary

then

( ) ( )∫+∞

∞−

−= tdetsfS tfi π2 ( ) ( )∫+∞

∞−

= fdefSts tfi π2

( ) ( )

( )

2/1

2

22

:

−

=∆

∫

∫∞+

∞−

+∞

∞−

tdts

tdtstt

t

( )

( )∫

∫∞+

∞−

+ ∞

∞−=tdts

tdtst

t2

2

:

Signal Duration Signal Median

( ) ( )

( )

2/1

2

2224

:

−

=∆

∫

∫∞+

∞−

+∞

∞−

fdfS

fdfSff

f

π ( )

( )∫

∫∞+

∞−

+ ∞

∞−=fdfS

fdfSf

f2

22

:

π

Signal Bandwidth Frequency Median

Fourier

( ) ( )ft ∆∆≤2

1


43


( ) ( ) ( )[ ]tttats θω += 0cos

SOLOComplex Representation of Bandpass Signals The majority of radar signals are narrow band signals, whose Fourier transform islimited to an angular-frequency bandwidth of W centered about a carrier angularfrequency of ±ω0.

Another form of s (t) is

( ) ( ) ( )( )

( ) ( ) ( )( )

( )

( ) ( ) ( ) ( )ttstts

tttatttats

QI

tsts QI

00

00

sincos

sinsincoscos

ωω

ωθωθ

−=

−=

sI (t) – in phase component sQ (t) – quadrature component

1

2

Define the signal complex envelope: ( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta

tjttatsjtstg QI

θθθ

exp

sincos:

=

+=+=

Therefore:

( ) ( ) ( )[ ] ( )[ ]tstjtgts ReexpRe 0 == ω

( ) ( ) ( ) ( ) ( ) ( ) ( )tststjtgtjtgts *2

1

2

1exp

2

1exp

2

100 +=−+= ∗ ωω

or:

3

4

( ) ( ) ( )[ ]tjtjtats θω += 0expAnalytic (complex) signal

44


( ) ( ) ( )[ ]tttats θω += 0cos

SOLOAutocorrelation The Autocorrelation Function is extensively used in Radar Signal Processing

( ) ( ) ( )∫+∞

∞−

−= tdtstsRss ττ :

Real signal For

The Autocorrelation Function is defined as:

Properties of the Autocorrelation Function:

2 ( ) ( )ττ ssss RR =−

( ) ( ) ( ) ( ) ( ) ( )τττττ

ss

tt

ss RtdtststdtstsR =−=+=− ∫∫+∞

∞−

+=+∞

∞−

''''

1 ( ) ( ) ( ) ( ) ( ) sss EfdfSfStdtstsR === ∫∫+∞

∞−

+∞

∞−

*0 Es – signal energy

3

( ) ( ) ( ) ( ) ( ) ( )2222

2

20sss

EE

InequalitySchwarz

ss REtdtstdtstdtstsR

ss

==−≤−= ∫∫∫∞+

∞−

∞+

∞−

∞+

∞−

τττ

( ) ( )0ssss RR ≤τ

Autocorrelation is a mathematical tool for finding specific patterns, such as the presence of a known signal which has been buried under noise.

45

Matched Filters for RADAR SignalsSOLOAutocorrelation (continue – 1(

The Autocorrelation Function is extensively used in Radar Signal Processing

( ) ( ) ( )∫+∞

∞−

−= tdtgtgRgg ττ *:

Signal complex envelope For


Properties of the Autocorrelation Function:

2 ( ) ( )ττ *gggg RR =−

( ) ( ) ( ) ( ) ( ) ( )τττττ

*''*'*'

gg

tt

gg RtdtgtgtdtgtgR =−=+=− ∫∫+∞

∞−

+=+∞

∞−

1 ( ) ( ) ( ) ( ) ( ) sgg EfdfGfGtdtgtgR 2**0 === ∫∫+∞

∞−

+∞

∞−

Es – signal energy

3

( ) ( ) ( ) ( ) ( ) ( )22

2

2

2

2

22

04** ggs

EE

InequalitySchwarz

gg REtdtgtdtgtdtgtgR

ss

==−≤−= ∫∫∫∞+

∞−

∞+

∞−

∞+

∞−

τττ

( ) ( )0gggg RR ≤τ

( ) ( ) ( )[ ]tjtatg θexp:=

46

Matched Filters for RADAR SignalsSOLOAutocorrelation (continue – 2(

The Autocorrelation Function is extensively used in Radar Signal Processing

( ) ( ) ( )∫+∞

∞−

−= tdtgtgRgg ττ *:

Signal complex envelope For


3

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

∫ ∫∫ ∫

∫ ∫∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

=

+∞

∞−

+∞

∞−

∂∂+

∂∂=

−−∂∂==

∂∂=

0

111222

2

0

222111

1

0

212211

2

****

**00

gggg RR

gg

tdtgtgtdtgt

tgtdtgtgtdtgt

tg

tdtdtgtgtgtgRτ

τττ

ττ

( ) ( )0gggg RR ≤τ

( ) ( ) ( )[ ]tjtatg θexp:=

(continue – 1)Since Rgg (0) is a maximum of a continuous function at τ=0, we must have

( ) 002

==∂∂ ττ ggR

Therefore ( ) ( ) ( ) ( ) 0** =∂∂+

∂∂

∫∫+∞

∞−

+∞

∞−

tdtgt

tgtdtgt

tg

47


( ) ( ) ( )[ ]tttats θω += 0cos

SOLO

Matched Filter for Received Radar Signals

The majority of radar signals are narrow band signals, whose Fourier transform islimited to an angular-frequency bandwidth of W centered about a carrier angularfrequency of ±ω0.

The received signal will be:

1

• attenuated by a factor α• retarded by a time t0 = 2 R/c

• affected by the Doppler effectc

RRc

f

c

D

222 0

2

00

ωλ

πωωπλ

−=−===

( ) ( ) ( ) ( ) ( )[ ]0000 cos ttttttats Dr −+−+−= θωωα2

Since the range and range-rate (t0, ωD) are not known exactly in advance, the matched filter is designed to match the received signal at any time t0

assuming zero Doppler ωD=0.Return to Table of Content

48


( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2

1exp

2

1 ωω −+= ∗

( ) ( )( ) ( )

≤≤−== −∗

Ttttsth

eSH tj

00

0ωωω

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]∫

∫∫∞+

∞−

∗∗

+∞

∞−

+∞

∞−

+−−+−++−+−

−+=

+−=−=

00000000

0

exp2

1exp

2

1exp

2

1exp

2

1ttjttgttjttgjgjg

dttssdthstso

τωττωττωττωτ

ττττττ

SOLO

The Matched Filter is a linear time-invariant filter hopt (t) that maximizesthe output signal-to-noise ratio at a predefined time t0, for a known transmitted signal s (t(.

Assuming no Doppler let find the Matched Filter for the received radar signal at a time t0:

( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2

1exp

2

1 ωω −+= ∗

( )[ ] ( ) ( ) ( )[ ] ( ) ( )∫∫+∞

∞−

∗+∞

∞−

∗ +−−−++−−= τττωτττω dttggttjdttggttj 000000 exp4

1exp

4

1

( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )∫∫+∞

∞−

∗+∞

∞−

∗ +−−−+−+−−+ τωττωτωττω dtjttggttjdtjttggttj 00000000 2expexp4

12expexp

4

1

Matched Filter Response to a Band Limited Radar Signal

49

Matched Filter output envelope


( ) ( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

+−=−= ττττττ dttssdthstso 0

SOLO

Matched Filter Response to a Band Limited Radar Signal (continue – 1(

The transmitted radar signal:

( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2

1exp

2

1 ωω −+= ∗

( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( )

−+−−+

+−−= ∫∫+∞

∞−

∗+∞

∞−

∗ τωττωτττω dtjttggttjdttggttj 0000000 2expexpRe2

1expRe

2

1

The integral in the second term on the r.h.s. is the Fourier transform ofevaluated at ω = 2 ω0. Since the spectrum of is limited by ω = W << ω0, thissecond term can be neglected, therefore:

( ) ( )[ ]0ttgg +−∗ ττ( )τg

( ) ( )[ ] ( ) ( ) ( ) ( )[ ]tjtgdttggttjts o

filtermatchedsignal

o 0000 expRe2

1expRe

2

1 ωτττω =

+−−≈ ∫∞+

∞−

∗

( ) [ ] ( ) ( ) [ ] ( )000000 exp2

1exp

2

1ttRtjdttggtjtg gg

filtermatchedsignal

o −−=+−−= ∫+∞

∞−

∗ ωτττω

( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta

tjttatsjtstg QI

θθθ

exp

sincos:

=

+=+=

Constant Phase

Matched Filter (for time t0) output is:

Autocorrelation Function of ( )tgReturn to Table of Content

50


Matched Filter Response to Phase Coding

( ) ( ) ( ) ∆<<

=∆−=∑−

= elsewhere

tttftptfctg

M

pp 0

011

0

Let the signal be a phase-modulated carrier, in which the modulation is in discrete and equal steps Δt. The complex envelope of the signal can be described by a sequence of complex numbers , such thatkc

( ) [ ] ( ) ( )∫+∞

∞−

∗ +−−= dtttgtgtjgo 000exp2

1 τωτ

Constant Phase

Matched Filter output envelope (change t ↔τ):

( )ttk ∆<≤+∆→ τττ 0

( ) [ ] ( ) ( )[ ]

[ ] ( )[ ]( )

∑ ∫

∫ ∑−

=

∆+

∆

∗

+∞

∞−

∗−

=

∆−+−∆−=

∆−+−∆−∆−=+∆

1

0

1

0

1

00

exp2

1

exp2

1

M

p

tp

tp

p

M

ppo

dttkMtgctMj

dttkMtgtptfctMjtkg

τω

τωτ

Change variable of integration to t1 = t – τ + (M - k) Δt

( ) [ ] ( )( )

( )

∑ ∫−

=

−∆+−+

−∆−+

∗∆−=+∆1

0

1

110exp2

1 M

p

tkMp

tkMp

po dttgctMjtkgτ

τ

ωτ

tMt ∆=0 (expected receiving time)

( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2

1exp

2

1 ωω −+= ∗The signal:

51


Matched Filter Response to Phase Coding (continue – 1(

Matched Filter output envelope for a Phase Coding is:

( ) [ ] ( )[ ]( )

∑ ∫−

=

∆+

∆

∗ ∆−+−∆−=+∆1

0

1

0exp2

1 M

p

tp

tp

po dttkMtgctMjtkg τωτ

Change variable of integration to t1 = t – τ + (M - k) Δt

( ) [ ] ( )( )

( )[ ] ( )

( )

( )( )

( )

( )

∑ ∫∫∑ ∫−

=

−∆+−+

∆−+

∗∆−+

−∆−+

∗−

=

−∆+−+

−∆−+

∗

+∆−=∆−=+∆

1

0

1

11110

1

0

1

110 exp2

1exp

2

1 M

p

tkMp

tkMp

tkMp

tkMp

p

M

p

tkMp

tkMp

po dttgdttgctMjdttgctMjtkgτ

τ

τ

τ

ωωτ

( ) ( ) ( )( ) ( ) ( ) τ

τ−∆+−+<<∆−+=

∆−+<<−∆−+=

−+∗

−−+∗

tkMpttkMpctg

tkMpttkMpctg

kMp

kMp

11*

1

11*

1

( ) [ ] ∑ ∫∫−

=

−∆

−+

−

−−+

+∆−=+∆

1

0 0

1*

0

11*

0exp2

1 M

p

t

kMpkMppo dtcdtcctMjtkgτ

τ

ωτ

( ) [ ] ∑−

=−+−−+

∆

−+

∆

∆−∆

=+∆1

0

*1

*0 1exp

2

1 M

p

kMpkMppo tc

tcctMj

ttkg

ττωτ

This equation describes straight lines in the complex plane, that can have corners only atτ = 0. At those corners

( ) [ ] ∑−

=−+∆−

∆=∆

1

0

*0exp

2

1 M

p

kMppo cctMjt

tkg ω

Constant Phase

52



Matched Filter output envelope for a Phase Coding is:

( ) [ ] ∑−

=−+−−+

∆

−+

∆

∆−∆

=+∆1

0

*1

*0 1exp

2

1 M

p

kMpkMppo tc

tcctMj

ttkg

ττωτ

This equation describes straight lines in the complex plane, that can have corners only atτ = 0. At those corners

( ) [ ] ∑−

=−+∆−

∆=∆

1

0

*0exp

2

1 M

p

kMppo cctMjt

tkg ω

Constant Phase

We can see that is the Discrete Autocorrelation Function for the observation time t0 = M Δt (the time the received Radar signal return is expected)

∑−

=−+

1

0

*M

p

kMpp cc

53



Example: Pulse poly-phase coded of length 4

Given the sequence: { } 1,,,1 −−++= jjck

which corresponds to the sequence of phases 0◦, 90◦, 270◦ and 180◦, the matched filter is given in Figure bellow.

{ } 1,,,1* −+−+= jjck

54

Pulse poly-phase coded of length 4

At the Receiver the coded pulse enters a 4 cells delay lane (from left to right), a bin at each clock.The signals in the cells are multiplied by -1,+j,-j or +1 and summed.

clock

SOLOPoly-Phase Modulation

-1 = -11 1+

-j +j = 02 1+j+

+j -1-j = -13 1+j+j−

+1 +1+1+1 = 44 1+j+j−1−

-j-1+j = -15 j+j−1−

+j - j = 06

j−1−7

1− -1 = -1

8 0

Σ

{ } 1,,,1 −−++= jjck

1− 1+j+ j− {ck*}

0 = 00

0

1

2

3

4

5

6

7

{ } 1,,,1* −+−+= jjck

55

-1

Pulse bi-phase Barker coded of length 3

Digital Correlation At the Receiver the coded pulse enters a 3 cells delay lane (from left to right), a bin at each clock.The signals in the cells are multiplied according to ck* sign and summed.

clock

-1 = -11

+1 -1 = 02

-( +1) = -15

0 = 06

+1 +1-( -1) = 33

+1-( +1) = 04

SOLO Pulse Compression Techniques

1

2

3

4

5

6

0

+1+1

0 = 00

56

Pulse bi-phase Barker coded of length 5

Digital CorrelationAt the Receiver the coded pulse enters a7 cells delay lane (from left to right),a bin at each clock.The signals in the cells are multipliedby ck* and summed.

clock

SOLO Pulse Compression Techniques

+1-1+1+1+1 { }*kc

+1 = +11

+1 = 19

0 = 010

2 -1 +1 = 0

+1 +1 -1-( +1) = 04

+1 +1 +1 –(-1)+1 = 55

0 = 0 0

3 +1-1 +1 = 1

+1 +1 -(+1) -1 = 06

+1-( +1) +1 = 17

–(+1) +1 = 08


57


( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta

tjttatsjtstg QI

θθθ

exp

sincos:

=

+=+=

SOLO

Matched Filter Response to its Doppler-Shifted Signal

Matched Filter for the transmitted radar signal:

The received radar signal has the form:

( ) ( ) ( )[ ]( ) ( ) ( ) ( )tjtgtjtg

tttats

00

0

exp2

1exp

2

1

cos

ωω

θω

−+=

+=

∗

( ) ( ) ( ) ( )[ ]000 cos tttttakts Dr −++−= θωω

( ) ( ) ( ) ( )[ ]

( ) ( )[ ] ( ) ( ) ( )[ ] ( )tjtjtgtjtjtgk

tttakts

DD

Dtr

0*

0

00

expexp2

1expexp

2

cos0

ωωωω

θωω

−+=

++==

( ) ( ) ( )∫+∞

∞−

∗=

−= τττ dtggtgfiltersignal

to 2

100

Matched Filter output envelope (designed under zero Doppler assumption) was found to be:

( ) ( ) ( ) ( )∫+∞

∞−

∗=

−= τττωτω dtgjgtgfiltersignal

DtDo exp

2

1,

00

For a nonzero Doppler (ωD ≠ 0) the Matched Filter output envelope is:

58


( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta

tjttatsjtstg QI

θθθ

exp

sincos:

=

+=+=

SOLO

Matched Filter Response to its Doppler-Shifted Signal (continue – 1(

For a nonzero Doppler (ωD ≠ 0) the Matched Filter output complex envelope is:

( ) ( ) ( ) ( )∫+∞

∞−

∗=

−= τττωτω dtgjgtgfiltersignal

DtDo exp

2

1,

00

Change between t and τ and define:

( ) ( ) ( ) ( )∫+∞

∞−

∗ −= dttfjtgtgfX DD πττ 2exp:,

The magnitude of the complex envelope ,is called the Ambiguity Function. ( )DfX ,τ

The name is sometimes used for , and sometimes even for . ( )DfX ,τ ( ) 2, DfX τ

59



Properties of: ( ) ( ) ( ) ( )∫+∞

∞−


( ) ( ) ( ) ( ) ( ) ( )∫∫∫+∞

∞−

∗+∞

∞−

+∞

∞−

∗ === dffGfGdttgdttgtgX2

:0,01

2 ( ) ( ) ( )DDD fXfjfX ,*2exp, ττπτ =−−( ) ( ) ( ) ( )

( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) [ ] ( ) ( )DDDD

tt

DD

DD

fXfjdttfjtgtgfj

dttfjtgtgfj

dttfjtgtgfX

,*2exp''2exp''*2exp

2exp2exp

2exp,

*'

ττππττπ

τπττπ

πττ

τ=

−=

+−+=

=−+=−−

∫

∫

∫

∞+

∞−

+=

∞+

∞−

∗

+∞

∞−

∗

( ) ( ) ( )DDD fXfjfX −−=− ,*2exp, ττπτ3

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( )[ ] ( ) ( )DDDD

tt

DD

DD

fXfjdttfjtgtgfj

dttfjtgtgfj

dttfjtgtgfX

−−=

−−−=

++−=

=+=−

∫

∫

∫

∞+

∞−

+=

∞+

∞−

∗

+∞

∞−

∗

,*2exp''2exp''*2exp

2exp2exp

2exp,

*'

ττππττπ

τπττπ

πττ

τ

60



Properties of: ( ) ( ) ( ) ( )∫+∞

∞−


4

5

( ) ( ) ( ) ( )τττ ggRdttgtgX =−= ∫+∞

∞−

∗ :0,

( ) ( ) ( ) ( ) ( ) ( ) ( )fRdfffGfGdttfjtgtgfX GGDDD =+== ∫∫+∞

∞−

+∞

∞−

∗ *2exp,0 π

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )[ ] ( ) ( ) ( )[ ]

( ) ( ) ( )fRdfffGfG

dfdttffjtgfGdfdttffjtgfG

dttfjtgdftfjfGdttfjtgtgfX

GGD

DD

DDD

=+=

+−=+=

==

∫

∫ ∫∫ ∫

∫ ∫∫

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∗

+∞

∞−

∗+∞

∞−

+∞

∞−

∗

*

*

2exp2exp

2exp2exp2exp,0

ππ

πππ


( ) [ ] ( ) ( ) [ ] ( ) [ ] ( )0,exp2

1exp

2

1exp

2

1000000000 ttXtjttRtjdttggtjtg gg

filtermatchedsignal

o −−=−−=+−−= ∫+∞

∞−

∗ ωωτττω

Autocorrelation Function of theSignal Complex Envelope ( )tg

We found that the Matched Filter Output Complex Envelope is:( )tgo

61

SOLO

Continue toAmbiguity Functions

Matched Filters and Ambiguity Functions for RADAR Signals

January 18, 2015 62

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 –2013

Stanford University1983 – 1986 PhD AA

63

Fourier Transform

( ) ( ){ } ( ) ( )∫+∞

∞−

−== dttjtftfF ωω exp:F

SOLO

Jean Baptiste JosephFourier

1768 - 1830

F (ω) is known as Fourier Integral or Fourier Transformand is in general complex

( ) ( ) ( ) ( ) ( )[ ]ωφωωωω jAFjFF expImRe =+=

Using the identities

( ) ( )tdtj δ

πωω =∫

+∞

∞− 2exp

we can find the Inverse Fourier Transform ( ) ( ){ }ωFtf -1F=

( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )[ ]002

1

2exp

2expexp

2exp

++−=−=−=

−=

∫∫ ∫

∫ ∫∫∞+

∞−

∞+

∞−

∞+

∞−

+∞

∞−

+∞

∞−

+∞

∞−

tftfdtfdd

tjf

dtjdjf

dtjF

ττδττπωτωτ

πωωττωτ

πωωω

( ) ( ){ } ( ) ( )∫+∞

∞−

==πωωωω

2exp:

dtjFFtf -1F

( ) ( ) ( ) ( )[ ]002

1 ++−=−∫+∞

∞−

tftfdtf ττδτ

If f (t) is continuous at t, i.e. f (t-0) = f (t+0)

This is true if (sufficient not necessary)f (t) and f ’ (t) are piecewise continue in every finite interval1

2 and converge, i.e. f (t) is absolute integrable in (-∞,∞)( )∫+∞

∞−

dttf

64

( )atf −-1F

F ( ) ( )ωω ajF −exp

Fourier TransformSOLO( )tf

-1FF ( )ωFProperties of Fourier Transform (Summary)

Linearity 1 ( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫+∞

∞−

F

Symmetry 2

( )tF-1F

F ( )ωπ −f2

Conjugate Functions3 ( )tf *

-1FF ( )ω−*F

Scaling4 ( )taf-1F

F

a

Fa

ω1

Derivatives5 ( ) ( )tftj n−-1F

F ( )ωω

Fd

dn

n

( )tftd

dn

n

-1FF ( ) ( )ωω Fj n

Convolution6

( ) ( )tftf 21-1F

F ( ) ( )ωω 21 * FF( ) ( ) ( ) ( )∫+∞

∞−

−= τττ dtfftftf 2121 :*-1F

F ( ) ( )ωω 21 FF

( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

= ωωω dFFdttftf 2*

12*

1

Parseval’s Formula7

Shifting: for any a real 8( ) ( )tajtf exp

-1FF ( )aF −ω

Modulation9 ( ) ttf 0cos ω-1F

F( ) ( )[ ]002

1 ωωωω −++ FF

( ) ( ) ( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

−=−= ωωωπ

ωωωπ

dFFdFFdttftf 212121 2

1

2

1

65

Fourier Transform

( )tf

( ) ( )∑∞

=

−=0n

T Tntt δδ

( ) ( ) ( ) ( ) ( )∑∞

=

−==0

*

n

T TntTnfttftf δδ

( )tf *

( )tfT t

( ) ( ){ } ( ) σσ <==+∫

∞

−f

ts dtetftfsF0

L

SOLO

Sampling and z-Transform

( ) ( ){ } ( ) σδδ <−

==

−==−

∞

=

−∞

=∑∑ 0

1

1

00sT

n

sTn

n

T eeTnttsS LL

( ) ( ){ }( ) ( ) ( )

( ) ( ){ } ( ) ( )

<<−

=

=

−

==

−

∞+

∞−−−

∞

=

−∞

=

+∫

∑∑

0

00**

1

1

2

1 σσσξξπ

δ

δ

ξ

σ

σξ f

j

j

tsT

n

sTn

n

de

Fj

ttf

eTnfTntTnf

tfsF

L

LL

( )

( ) ( )( )

( )( )

( )

( )

( )( )

( )( )

( )

−=

−

−=

−=

∑∫

∑∫

∑

−−−

−−

Γ

−−

−−

Γ

−−

∞

=

−

tse

ofPoleststs

FofPoles

tsts

n

nsT

e

FResd

e

F

j

e

FResd

e

F

j

eTnf

sF

ξ

ξξ

ξ

ξξ

ξξξπ

ξξξπ

1

1

0

*

112

1

112

1

2

1

Poles of

( ) Tse ξ−−−1

1

Poles of

( )ξF

planes

Tnsn

πξ 2+=

ωj

ωσ j+

0=s

Laplace Transforms

The signal f (t) is sampled at a time period T.

1Γ2Γ

∞→R

∞→R

Poles of

( ) Tse ξ−−−1

1

Poles of

( )ξF

planeξ

Tnsn

πξ 2+=

ωj

ωσ j+

0=s

66

Fourier Transform

( )tf

( ) ( )∑∞

=

−=0n

T Tntt δδ

( ) ( ) ( ) ( ) ( )∑∞

=

−==0

*

n

T TntTnfttftf δδ

( )tf *

( )tfT t

SOLO

Sampling and z-Transform (continue – 1)

( ) ( )( )

( )

( )

( ) ( ) ∑∑

∑∑

∞+

−∞=

∞+

−∞=−−→

∞+

−∞=−−

+→

+=−

−−

+=

−

+

−=

+

−

−−−=

−−=

−−

−−

nnTse

nts

T

njs

T

njs

e

ofPolests

T

njsF

TeT

Tn

jsF

T

njsF

eT

njs

e

FRessF

ts

n

ts

ππ

ππξξ

ξ

ξπξ

πξ

ξ

ξ

ξ

212

lim

2

1

2

lim1

1

2

21

1

*

Poles of

( )ξF

ωj

σ0=s

T

π2

T

π2

T

π2

Poles of

( )ξ*F plane

js ωσ +=


The poles of are given by( ) tse ξ−−−1

1

( ) ( )T

njsnjTsee n

njTs πξπξπξ 221 2 +=⇒=−−⇒==−−

( ) ∑+∞

−∞=

+=

n T

njsF

TsF

π21*

67

Fourier Transform

( )tf

( ) ( )∑∞

=

−=0n

T Tntt δδ

( ) ( ) ( ) ( ) ( )∑∞

=

−==0

*

n

T TntTnfttftf δδ

( )tf *

( )tfT t

SOLO


0=z

planez

Poles of

( )zF

C


The z-Transform is defined as:

( ){ } ( ) ( )( )

( ) ( )( )

−

−===

∑

∑

=

−

→

∞

=

−

=

iF

iF

iiF

Ts

FofPoles

T

F

n

n

ze

ze

F

zTnf

zFsFtf

ξξξ

ξ

ξξξξξ

1

0*

1

lim:Z

( ) ( )

<

>≥= ∫ −

00

02

1 1

n

RzndzzzFjTnf

fCC

n

π

68

Fourier TransformSOLO


( ) ( ) ( )∑∑∞

=

−+∞

−∞=

=

+=

0

* 21

n

nsT

n

eTnfT

njsF

TsF

πWe found

The δ (t) function we have:

( ) 1=∫+∞

∞−

dttδ ( ) ( ) ( )τδτ fdtttf =−∫+∞

∞−

The following series is a periodic function: ( ) ( )∑ −=n

Tnttd δ:

therefore it can be developed in a Fourier series:

( ) ( ) ∑∑

−=−=

n

n

n T

tnjCTnttd πδ 2exp:

where: ( )T

dtT

tnjt

TC

T

T

n

12exp

12/

2/

=

= ∫

+

−

πδ

Therefore we obtain the following identity:

( )∑∑ −=

−

nn

TntTT

tnj δπ2exp

Second Way

69

Fourier Transform

( ) ( ){ } ( ) ( )∫+∞

∞−

−== dttjtftfF νπνπ 2exp:2 F

( ) ( ) ( )∑∑∞

=

−+∞

−∞=

=

+=

0

* 21

n

nsT

n

eTnfT

njsF

TsF

π

( ) ( ){ } ( ) ( )∫+∞

∞−

== ννπνπνπ dtjFFtf 2exp2:2-1F

SOLOSampling and z-Transform (continue – 4)

We found

Using the definition of the Fourier Transform and it’s inverse:

we obtain ( ) ( ) ( )∫+∞

∞−

= ννπνπ dTnjFTnf 2exp2

( ) ( ) ( ) ( ) ( ) ( )∑∫∑∞

=

+∞

∞−

∞

=

−=−=0

111

0

* exp2exp2expnn

n sTndTnjFsTTnfsF ννπνπ

( ) ( ) ( )[ ]∫ ∑+∞

∞−

+∞

−∞=

−−== 111

* 2exp22 νννπνπνπ dTnjFjsFn

( ) ( ) ∑∫ ∑+∞

−∞=

+∞

∞−

+∞

−∞=

−=

−−==

nn T

nF

Td

T

n

TFjsF νπνννδνπνπ 2

1122 111

*

We recovered (with –n instead of n) ( ) ∑+∞

−∞=

+=

n T

njsF

TsF

π21*

Second Way (continue)

Making use of the identity: with 1/T instead of T

and ν - ν 1 instead of t we obtain: ( )[ ] ∑∑

−−=−−

nn T

n

TTnj 11

12exp ννδννπ

( )∑∑ −=

−

nn

TntTT

tnj δπ2exp

70

Fourier TransformSOLO

Henry Nyquist1889 - 1976

http://en.wikipedia.org/wiki/Harry_Nyquist

Nyquist-Shannon Sampling Theorem

Claude Elwood Shannon 1916 – 2001

http://en.wikipedia.org/wiki/Claude_E._Shannon

The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result, and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step response Integralsinus; this band-limiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English).

The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). V. A. Kotelnikov published similar results in 1933 ("On the transmission capacity of the 'ether' and of cables in electrical communications", translation from the Russian), as did the mathematician E. T. Whittaker in 1915 ("Expansions of the Interpolation-Theory", "Theorie der Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function theory"), and Gabor in 1946 ("Theory of communication").

http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem

71

SignalsSOLO


then

( ) ( )∫+∞

∞−

−= tdetsfS tfi π2 ( ) ( )∫+∞

∞−

= fdefSts tfi π2

t

t∆2

t

( ) 2ts

ff

f∆2

( ) 2fS

( ) ( )

( )

2/1

2

22

:

−

=∆

∫

∫∞+

∞−

+∞

∞−

tdts

tdtstt

t

( )

( )∫

∫∞+

∞−

+ ∞

∞−=tdts

tdtst

t2

2

:

Signal Duration Signal Median

( ) ( )

( )

2/1

2

2224

:

−

=∆

∫

∫∞+

∞−

+∞

∞−

fdfS

fdfSff

f

π ( )

( )∫

∫∞+

∞−

+ ∞

∞−=fdfS

fdfSf

f2

22

:

π

Signal Bandwidth Frequency Median

Fourier

72

Signals

( ) ( )∫+∞

∞−

= fdefSts tfi π2

SOLO


( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∫∫ ∫

∫ ∫∫ ∫∫∞+

∞−

∞+

∞−

∞+

∞−

−

∞+

∞−

∞+

∞−

−∞+

∞−

∞+

∞−

∞+

∞−

=

=

=

=

dffSfSdfdesfS

dfdefSsdfdefSsdss

tfi

tfitfi

ττ

τττττττ

π

ππ

2

22

( ) ( )∫+∞

∞−

= fdefSts tfi π2 ( ) ( ) ( )∫+∞

∞−

== fdefSfitd

tsdts tfi ππ 22'

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )∫∫ ∫

∫ ∫∫ ∫∫∞+

∞−

∞+

∞−

∞+

∞−

−

+∞

∞−

+∞

∞−

−+∞

∞−

+∞

∞−

−+∞

∞−

=

−=

−=

−=

dffSfSfdfdesfSfi

dfdesfSfidfdefSfsidss

tfi

tfitfi

222

22

2'2

'2'2''

πττπ

ττπττπτττ

π

ππ

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSds 22 ττ

Parseval Theorem

From

From

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSfdtts2222

4' π

73

Signals

( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )∫

∫

∫

∫ ∫

∫

∫ ∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−

−

∞+

∞−

+∞

∞−

+∞

∞−

−

∞+

∞−

+∞

∞−∞+

∞−

+∞

∞− =====dffS

fdfdfSd

fSi

dffS

fdtdetstfS

dffS

tdfdefStst

dffS

tdtstst

tdts

tdtst

t

fifi

22

2

2

2

22

2

2:

πππ

SOLO


( ) ( )∫+∞

∞−

−= tdetsfS tfi π2 ( ) ( )∫+∞

∞−

= fdefSts tfi π2Fourier

( ) ( )∫+∞

∞−

−−= tdetstifd

fSd tfi ππ 22( ) ( )∫

+∞

∞−

= fdefSfitd

tsd tfi ππ 22

( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )∫

∫

∫

∫ ∫

∫

∫ ∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−∞+

∞−

+∞

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−

−=

====tdts

tdtd

tsdtsi

tdts

tdfdefSfts

tdts

fdtdetsfSf

tdts

fdfSfSf

fdfS

fdfSf

f

fifi

22

2

2

2

22

2 2222

:

ππ ππππ

74

Signals

( ) ( ) ( ) ( ) ( )∫∫∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

+∞

∞−

=≤

dffSfdttstdttsdttstdtts

222222

2

2 4'4

1 π

( ) ( )∫∫+∞

∞−

+∞

∞−

= dffSdts22 τ

SOLO



From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−


Choose ( ) ( ) ( ) ( ) ( )tstd

tsdtgtsttf ':& ===

( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

≤ dttsdttstdttstst22

''we obtain

( ) ( )∫+∞

∞−

dttstst 'Integrate by parts( )

=+=

→

==

sv

dtstsdu

dtsdv

stu '

'

( ) ( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

∞+

∞−

+∞

∞−

−−= dttststdttsstdttstst '' 2

0

2

( ) ( ) ( )∫∫

+∞

∞−

+∞

∞−

−= dttsdttstst 2

2

1'

( ) ( )∫∫+∞

∞−

+ ∞

∞−

= dffSfdtts2222

4' π

( )

( )

( )

( )

( )

( )

( )

( )∫

∫

∫

∫

∫

∫

∫

∫∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞−∞+

∞−

+∞

∞− =≤dffS

dffSf

dtts

dttst

dtts

dffSf

dtts

dttst

2

222

2

2

2

222

2

244

4

1ππ

assume ( ) 0lim =→∞

tstt

75

SignalsSOLO


( )

( )

( )

( )

( )

( )

22

2

222

2

24

4

1

ft

dffS

dffSf

dtts

dttst

∆

∞+

∞−

+∞

∞−

∆

∞+

∞−

+∞

∞−

≤

∫

∫

∫

∫ π

Finally we obtain ( ) ( )ft ∆∆≤2

1


Since Schwarz Inequality: becomes an equalityif and only if g (t) = k f (t), then for:

( ) ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−


( ) ( ) ( ) ( )tftsteAttd

sdtgeAts tt ααα αα 222:

22

−=−=−==⇒= −−

we have ( ) ( )ft ∆∆=2

1

76

SOLO

77

SOLO

78

SOLO

4 matched filters and ambiguity functions for radar signals

Science